Gravesande, Willem Jacob 's - An essay on perspective

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7334An ESSAY the Circular Baſe of the Cone will be deter-
min’d.

Demonstration.

To prove this, draw the Lines B C and L F, cut-
ting the Line A S in the Points N and M; and make
the Line G n equal to A N, and draw the Line
n D m. It is now manifeſt, that if the Cone be
continued out above its Vertex, (that is, if the oppo-
ſite Cone be form’d) it will cut the Horizontal Plane
in a Circle equal to B E C, whoſe Seat will be BEC:
So that the Point S, in reſpect of B E C, is in the
ſame Situation as the Eye hath, with reſpect to the
Circle form’d in the Horizontal Plane, by the Conti-
nuation of the Cone. Whence it follows, that B C
is the Seat of the viſible Portion of that Circle. For,
by Conſtruction, B and C are the Points of Contact
of the Tangents to the Circle B E C, which paſs
thro’ the Point S; becauſe the Angle ABS, which
is in a Semicircle, is a right one.

Now, if a Plane be conceiv’d, as paſſing thro’ ſome
Points in the Horizontal Plane, whoſe Seats are
B and C, and which cuts the two oppoſite Cones
thro’ their Vertex; it is evident, that this Plane
continued, will cut the Geometrical Plane in a Line
parallel to B N C; and that this Line upon the
ſaid Plane, will determine the viſible Part of the
Cone’s Baſe. So, ſince G n was made equal to
A N, we have only to prove, that P m is equal to
A M: For, it follows from thence, that L M F is
the Common Section of the Geometrical Plane, and
the Plane which we have here imagin’d.

The Triangles D Q P and G H D are ſimilar, whence
D G: D P: : G H: P Q.

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Scan	Original
71	32
72	33
73	34
74	35
75	36
76
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79
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85	37
86	38
87	39
88	40
89	41
90	42
91
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94	43
95	44
96	45
97	46
98
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Scan	Original
71	32
72	33
73	34
74	35
75	36
76
77
78
79
80
81
82
83
84
85	37
86	38
87	39
88	40
89	41
90	42
91
92
93
94	43
95	44
96	45
97	46
98
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100

Scan	Original
71	32
72	33
73	34
74	35
75	36
76
77
78
79
80
81
82
83
84
85	37
86	38
87	39
88	40
89	41
90	42
91
92
93
94	43
95	44
96	45
97	46
98
99
100